Question 1177857: Suppose that the sitting back-to-knee length for a group of adults has a normal distribution with a mean of mu=22.9in. and a standard deviation of alpha= 1.2in. These data are often used in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. Instead of using 0.05 for identifying significant values, use the criteria that a value x is significantly high if P(x or greater)0.01 and a value is significantly low if P(x or less)0.01. Find the back-to-knee lengths separating significant values from those that are not significant. Using these criteria, is a back-to-knee length of 25.2in. significantly high?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**1. Find the Z-scores for the Significant Values**
* We are given that a value is significantly low if P(x or less) < 0.01.
* We are given that a value is significantly high if P(x or greater) < 0.01.
To find the back-to-knee lengths that separate significant values, we need to find the z-scores corresponding to the 0.01 and 0.99 percentiles of the standard normal distribution.
* For the lower bound (0.01 percentile), we need to find the z-score such that P(Z < z) = 0.01.
* Using a standard normal distribution table or a calculator, we find that z_lower ≈ -2.33.
* For the upper bound (0.99 percentile), we need to find the z-score such that P(Z > z) = 0.01, or equivalently, P(Z < z) = 0.99.
* Using a standard normal distribution table or a calculator, we find that z_upper ≈ 2.33.
**2. Calculate the Back-to-Knee Lengths**
We are given:
* Mean (μ) = 22.9 inches
* Standard deviation (σ) = 1.2 inches
We use the formula:
* x = μ + z * σ
* Lower bound:
* x_lower = 22.9 + (-2.33) * 1.2
* x_lower = 22.9 - 2.796
* x_lower ≈ 20.104 inches
* Upper bound:
* x_upper = 22.9 + (2.33) * 1.2
* x_upper = 22.9 + 2.796
* x_upper ≈ 25.696 inches
So, the back-to-knee lengths separating significant values are approximately 20.10 inches (lower bound) and 25.70 inches (upper bound).
**3. Determine if 25.2 Inches is Significantly High**
* We are asked if a back-to-knee length of 25.2 inches is significantly high.
* We found that the upper bound for significantly high values is approximately 25.70 inches.
* Since 25.2 inches < 25.70 inches, a back-to-knee length of 25.2 inches is not significantly high.
**Results**
* Lower bound for significant values: ≈ 20.10 inches
* Upper bound for significant values: ≈ 25.70 inches
* A back-to-knee length of 25.2 inches is not significantly high.
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